% \Chapter{Fixed-point-free automorphism groups} % The functions described in this chapter are purely group-theoretic and are meant to provide solvable fixed-point-free automorphism groups acting on abelian groups (Frobenius groups with abelian Frobenius kernel and solvable Frobenius complement) for the construction of centralizer nearrings, planar nearrings, designs, and so on. The classification of fixed-point-free automorphism groups in types I - IV follows Zassenhaus' papers and \cite{Wolf:Spaces}. %J. A. Wolf, Spaces of Constant curvature. The fixed-point-free automorphism groups acting on abelian groups are constructed from fixed-point-free representations as described in \cite{Mayr:Representations}. %Peter Mayr, Fixed-point-free Representations over Fields of Prime Characteristic, Johannes Kepler University Linz - Reports of the Mathematical Institutes, 554, 2000. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Fixed-point-free automorphism groups and Frobenius groups} \>IsFpfAutomorphismGroup( <phi>, <G> ) An automorphism group $\Phi$ of a group $G$ acts fixed-point-free (fpf) on $G$ if $\Phi$ has more than 1 element and no automorphism in $\Phi$ except the identity mapping has a fixed point besides the group identity of $G$. $\Phi$ is fpf on $G$, iff the semidirect product of $G$ and $\Phi$, with $\Phi$ acting naturally on $G$, is a Frobenius group. The function `IsFpfAutomorphismGroup' returns the according value `true' or `false' for a group of automorphisms <phi> on the group <G>. \beginexample gap> C9 := CyclicGroup( 9 ); <pc group of size 9 with 2 generators> gap> a := GroupHomomorphismByFunction( C9, C9, x -> x^-1 );; gap> phi := Group( a );; gap> Size( phi ); 2 gap> IsFpfAutomorphismGroup( phi, C9 ); true \endexample \>FpfAutomorphismGroupsMaxSize( <G> ) `FpfAutomorphismGroupsMaxSize' returns a list with integers <kmax> and <dmax> where <kmax> is an upper bound for the size of an fpf automorphism group on the group <G>; for example, the order of <G> is congruent to 1 modulo <kmax> and <kmax> is odd for nonabelian groups <G>. The order of any fpf automorphism group <phi> on <G> divides <kmax>. Let <phi> be a metacyclic fpf automorphism group acting on <G>. Then <phi> has a cyclic normal subgroup whose index in <phi> divides <dmax>. Thus, if <dmax> is 1, then <G> admits cyclic fpf automorphism groups only. \beginexample gap> G := ElementaryAbelianGroup( 49 );; gap> FpfAutomorphismGroupsMaxSize( G ); [ 48, 2 ] gap> C15 := CyclicGroup( 15 );; gap> FpfAutomorphismGroupsMaxSize( C15 ); [ 2, 1 ] \endexample \>FrobeniusGroup( <phi>, <G> ) `FrobeniusGroup' constructs the semidirect product of <G> with the fpf automorphism group <phi> of <G> with the multiplication $(a,g)*(b,h)=(ab,g^ah)$ by using the function `SemidirectProduct'. Thus a Frobenius group with Frobenius kernel <G> and Frobenius complement <phi> where the action of <phi> on <G> is the natural action of automorphisms on the group is returned. The unique Frobenius group with kernel $G = (Z_{3})^2\times(Z_{5})^2$ and quaternion complement is constructed as follows: \beginexample gap> aux := FpfAutomorphismGroupsMetacyclic( [3,3,5,5], 4, -1 ); [ [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ], [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ], <pc group of size 225 with 4 generators> ] gap> phi := Group( aux[1][1] ); <group with 2 generators> gap> G := aux[2]; <pc group of size 225 with 4 generators> gap> FrobeniusGroup( phi, G ); <pc group of size 1800 with 7 generators> \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Fixed-point-free representations} \>IsFpfRepresentation( <matrices>, <F> ) Let $\pi$ be a representation of the group $\Phi$ over the finite field $F$. If for all $\varphi\in\Phi$ except for the identity the matrix $\pi(\varphi)$ does not have $1$ as an eigenvalue, then $\pi$ is said to be fpf. Let $\pi$ be an fpf representation of $\Phi$ over $F$ with degree $d$. Then $\pi$ is faithful, the order of $\Phi$ and the characteristic of $F$ are coprime and $\pi$ is a sum of irreducible faithful fpf $F$-representations. The matrix group $\pi(\Phi)$ acts fpf on the vectorspace $F^d$. For a list of $d\times d$ matrices, <matrices>, over the field <F>, the function `IsFpfRepresentation' returns `true' if the group generated by <matrices> acts fpf on the $d$-dimensional vectorspace over <F>, and `false' otherwise. \beginexample gap> F := GF(5);; gap> A := [[2,0],[0,3]]*One(F); [ [ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ] ] gap> IsFpfRepresentation( [A], F ); true \endexample \>DegreeOfIrredFpfRepCyclic( <p>, <m> ) returns the degree of the irreducible fpf representations of the cyclic group of order <m> over GF(<p>), where <m> and <p> are coprime. Note, that all irreducible fpf representations of the cyclic group of order <m> over GF(<p>) have the same degree, the multiplicative order of <p> modulo <m>, `OrderMod( p, m )'. \beginexample gap> DegreeOfIrredFpfRepCyclic( 5, 9 ); 6 \endexample \>DegreeOfIrredFpfRepMetacyclic( <p>, <m>, <r> ) returns the degree of the irreducible fpf representations of the metacyclic group $\Phi$ determined by parameters <m> and <r> over GF(<p>). If the parameters are not feasible, then an error is returned. See `FpfRepresentationsMetacyclic' for a presentation of this group. All irreducible fpf representations of the metacyclic group over GF(<p>) have the same degree, namely the size of multiplicative group generated by <p> and <r> modulo <m>. We determine the degree of the irreducible fpf representation of the quaternion group over GF(5): \beginexample gap> DegreeOfIrredFpfRepMetacyclic( 5, 4, -1 ); 2 \endexample \>DegreeOfIrredFpfRep2( <p>, <m>, <r>, <k> ) returns the degree of the irreducible fpf representations of the type-II-group $\Phi$ determined by parameters <m>, <r>, and <k> over GF(<p>). If the parameters are not feasible or if the parameters describe the presentation of a metacyclic group, then an error is returned. See `FpfRepresentations2' for a presentation of this group. All irreducible fpf representations of $\Phi$ over GF(<p>) have the same degree, namely the size of the multiplicative group generated by <p>, <r>, and <k> modulo <m>. We determine the degree of the irreducible fpf representation of the smallest, not metacyclic type-2-group (order 120) over the field GF(7): \beginexample gap> DegreeOfIrredFpfRep2( 7, 30, 11, -1 ); 8 \endexample \>DegreeOfIrredFpfRep3( <p>, <m>, <r> ) returns the degree of the irreducible fpf representations of the type-III-group $\Phi$ determined by parameters <m> and <r> over GF(<p>). If the parameters are not feasible, then an error is returned. See `FpfRepresentations3' for a presentation of this group. All irreducible fpf representations of this group over GF(<p>) have the same degree. We determine the degree of the irreducible fpf representation of SL(2,3) over GF(5): \beginexample gap> DegreeOfIrredFpfRep3( 5, 3, 1 ); 2 \endexample \>DegreeOfIrredFpfRep4( <p>, <m>, <r>, <k> ) returns the degree of the irreducible fpf representations of the type-IV-group $\Phi$ determined by parameters <m>, <r>, and <k> over GF(<p>). If the parameters are not feasible, then an error is returned. See `FpfRepresentations4' for a presentation of this group. All irreducible fpf representations of $\Phi$ over GF(<p>) have the same degree. We determine the degree of the irreducible fpf representation of the smallest type-4-group, the binary octahedral group of order 48, over $GF(5)$: \beginexample gap> DegreeOfIrredFpfRep4( 5, 3, 1, -1 ); 4 \endexample \>FpfRepresentationsCyclic( <p>, <m> ) Let $a$ generate a cyclic group of order <m>. %determines the nonequivalent irreducible fpf representations of a %cyclic group of order <m> over GF(<p>). For <p> and <m> coprime `FpfRepresentationsCyclic' returns a list of matrices $\{ A^i | i$ in $indexlist \}$ over GF(<p>) as well as the list $indexlist$. For all $i$ in $indexlist$ the representation $a \mapsto A^i$ is irreducible and fpf. The $A^i$ with $i$ in $indexlist$ describe all irreducible fpf representations up to equivalence; each irreducible fpf representation is equivalent to one $a \mapsto A^i$ and no two representations $a \mapsto A^i$, $a \mapsto A^j$ with $i\neq j$ and $i,j$ in $indexlist$ are equivalent. Note, that every faithful irreducible representation of a cyclic group is fpf. The number of nonequivalent faithful irreducible representations over GF(<p>) is given as $\phi(m)/d$, where the degree $d$ is given as the multiplicative order of <p> modulo <m> and $\phi(m)$ denotes the number of residues coprime to $m$. We determine the irreducible matrix representations of the cyclic group of size 8 over $GF(5)$: \beginexample gap> aux := FpfRepresentationsCyclic( 5, 8 ); [ [ [ [ Z(5)^3, Z(5)^2 ], [ Z(5), Z(5) ] ], [ [ Z(5)^2, Z(5) ], [ Z(5)^0, Z(5)^0 ] ] ], [ 1, 7 ] ] gap> mats := aux[1]; [ [ [ Z(5)^3, Z(5)^2 ], [ Z(5), Z(5) ] ], [ [ Z(5)^2, Z(5) ], [ Z(5)^0, Z(5)^0 ] ] ] gap> indexlist := aux[2]; [ 1, 7 ] \endexample \>FpfRepresentationsMetacyclic( <p>, <m>, <r> ) Let $\Phi$ be a metacyclic group (i.e., $\Phi$ has a cyclic normal subgroup with cyclic factor) admitting an fpf representation. Then $\Phi$ fulfills one of the following two presentations, I or II. Both presentations are determined by integers $m$ and $r$ satisfying certain conditions: Type I. Presentation of an fpf metacyclic group $\Phi$ with all Sylow subgroups cyclic. Let $m$ and $r$ satisfy the following conditions: \beginlist \item{(a)} $m$ and $r$ are coprime. \item{(b)} Let $n$ be the multiplicative order of $r$ modulo $m$. Then each prime divisor of $n$ divides $m$. \item{(c)} Let $m'$ be maximal such that $m'$ divides $m$ and $m'$ is coprime to $n$. Then $r = 1$ mod $(m/m')$. \endlist Type II. Presentation of an fpf metacyclic group $\Phi$ with generalized quaternion 2-Sylow subgroup. Let $m$ and $r$ satisfy the following conditions: \beginlist \item{(a)} $m$ and $r$ are coprime. \item{(b)} Let $n$ be the multiplicative order of $r$ modulo $m$. Then $n$ is $2$ times an odd number and each prime divisor of $n$ divides $m$. \item{(c)} Let $s$ be maximal such that $2^s$ divides $m$. Then $2^s\geq 4$ and $r = -1$ mod $2^s$. \item{(d)} Let $m'$ be maximal such that $m'$ divides $m/2$ and $m'$ is coprime to $n/2$. Then $r = 1$ mod $(m/m')$. \endlist Then the group $\Phi$ with 2 generators $a,b$ satisfying the relations $a^m = 1, b^n = a^{m'}, b^{-1}ab = a^r$ is metacyclic and fpf and has size $mn$. A group satisfying presentation I is of type I in the notation of Zassenhaus, Wolf; presentation II gives a type-II-group. Let <m>, <r> be as above, and let <p> coprime to <m>. Additionally, we require that <m> does not divide <r>-1. (Otherwise, $\Phi = \langle a,b | a^m = 1, b^n = a^{m'}, b^{-1}ab = a^r \rangle$ would be cyclic.) Then `FpfRepresentationsMetacyclic' returns a list of matrices $\{ (A^i,B_i) | i$ in $indexlist \}$ over GF(<p>) as well as the list $indexlist$. The GF(<p>)-representations determined by $a \mapsto A^i$ and $b \mapsto B_i$ are all irreducible and fpf representations of $\Phi = \langle a,b | a^m = 1, b^n = a^{m'}, b^{-1}ab = a^r \rangle$ up to equivalence. We determine the irreducible matrix representation of the quaternion group (parameters $m = 4, r = -1$) over $GF(7)$: \beginexample gap> aux := FpfRepresentationsMetacyclic( 7, 4, -1 ); [ [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ], [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ] ] ], [ 1 ] ] gap> mats := aux[1]; [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ], [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ] ] ] \endexample \>FpfRepresentations2( <p>, <m>, <r>, <k> ) The presentation of a type-II-group which is not metacyclic is determined by integers $m,r,k$ satisfying the following conditions: % (again in \cite{Wolf:Spaces}, 6.1.11, a slightly different notation is used): \beginlist \item{(a)} $m$ and $r$ are coprime, $m$ and $k$ are coprime. \item{(b)} Let $n$ be the multiplicative order of $r$ modulo $m$. Then $n$ is $2$ times an odd number and each prime divisor of $n$ divides $m$. \item{(c)} Let $m'$ be maximal such that $m'$ divides $m$ and $m'$ is coprime to $n$. Then $r = 1$ mod $(m/m')$. \item{(d)} Let $2^{s-1}$ be maximal such that $2^{s-1}$ divides $m$. Define $l = -1$ mod $2^{s-1}$ and $l = 1$ mod $(nm/(2^{s-1}m'))$. Then $k = l$ mod $(m/m')$. \item{(e)} The multiplicative order of $k$ modulo $m$ equals $2$ and $k\neq r^{(n/2)}$ mod $m$. \endlist Then the group $\Phi$ with generators $a,b,q$ satisfying the relations $a^m = 1, b^n = a^{m'}, b^{-1}ab = a^r$ and furthermore $q^{-1}a q = a^k, q^{-1}b q = b^l$ is fpf of type II and has size $2mn$. $a,b$ generate a metacyclic group with all Sylow subgroups cyclic (see conditions (a), (b), (c)) of index $2$ in $\Phi$. For <m>, <r>, <k> as above and <p> coprime to <m> `FpfRepresentations2' returns a list of matrices $\{ (A_i,B_i,Q_i) | i$ in $indexlist \}$ over GF(<p>) as well as the list $indexlist$. The GF(<p>)-representations determined by $a \mapsto A_i, b \mapsto B_i$ and $q \mapsto Q_i$ are all irreducible, fpf representations of $\Phi$ upto equivalence. We determine the irreducible matrix representations of the smallest type-II-group which is not metacyclic (parameters m = 30, r = 11, k = -1, size 120) over the field GF(11) and obtain 2 nonequivalent fpf representations, each of degree 4: \beginexample gap> DegreeOfIrredFpfRep2( 11, 30, 11, -1 ); 4 gap> aux := FpfRepresentations2( 11, 30, 11, -1 ); [ [ [ <block matrix of dimensions (2*2)x(2*2)>, <block matrix of dimensions (2*2)x(2*2)>, <block matrix of dimensions (2*2)x(2*2)> ], [ <block matrix of dimensions (2*2)x(2*2)>, <block matrix of dimensions (2*2)x(2*2)>, <block matrix of dimensions (2*2)x(2*2)> ] ], [ 1, 13 ] ] \endexample \>FpfRepresentations3( <p>, <m>, <r> ) A group $\Phi$ admitting an fpf representation is said to be of type III if $\Phi$ is the semidirect product of the quaternion group and a metacyclic fpf group $H$ of odd size, with the quaternion group normal and $H$ permuting the $3$ subgroups of order $4$. The presentation of a type-III-group is determined by integers $m$ and $r$, describing the metacyclic group $H$ and its action on the normal quaternion subgroup. The following conditions have to be satisfied for $m,r$: \beginlist \item{(a)} $3$ divides $m$; $m$ is odd; $m$ and $r$ are coprime. \item{(b)} Let $n$ be the multiplicative order of $r$ modulo $m$. Then each prime divisor of $n$ divides $m$. \item{(c)} Let $m'$ be maximal such that $m'$ divides $m$ and $m'$ is coprime to $n$. Then $r = 1$ mod $(m/m')$. \endlist Let $p,q$ with relations $p^4 = 1, q^2 = p^2, q^{-1}pq = p^{-1}$ generate the quaternion group. Let $a,b$ generate a metacyclic group determined by $m$ and $r$ (See `FpfRepresentationsMetacyclic'). If $3$ divides $n$, then let $a$ commute with $p,q$ and let $b^{-1}pb = q, b^{-1}qb = pq$. If $3$ does not divide $n$, then let $b$ commute with $p,q$ and let $a^{-1}pa = q, a^{-1}qa = pq$ Then the group $\Phi$ with generators $p,q,a,b$ is of type III and has size $8mn$. For $r \neq 1$ mod $m$, `FpfRepresentations3' returns a list of matrices $\{ (P, Q, A_i,B_i) | i$ in $indexlist \}$ over GF(<p>) as well as the list $indexlist$. For $r = 1$ mod $m$, the group $H$ is cyclic and `FpfRepresentations3' returns $\{ (P, Q, A_i) | i$ in $indexlist \}$ over GF(<p>) and $indexlist$. The GF(<p>)-representations determined by $p \mapsto P, q \mapsto Q$ and $a \mapsto A_i, b \mapsto B_i$ are all irreducible, fpf representations of $\Phi$ upto equivalence. We determine the irreducible matrix representation of the smallest type-III-group, namely SL(2,3), (parameters m = 3, r = 1, size 24) over the field GF(5): \beginexample gap> aux := FpfRepresentations3( 5, 3, 1 ); [ [ [ [ [ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ] ], [ [ 0*Z(5), Z(5)^2 ], [ Z(5)^0, 0*Z(5) ] ], [ [ Z(5)^3, Z(5)^0 ], [ Z(5), Z(5)^0 ] ] ] ], [ 1 ] ] \endexample \>FpfRepresentations4( <p>, <m>, <r>, <k> ) A group $\Phi = \langle p,q,a,b,z\rangle$ admitting an fpf representation is said to be of type IV, if it has a normal subgroup $H = \langle p,q,a,b\rangle$ of type III and index 2. The presentation of a type-IV-group is determined by integers $m,r,k$ satisfying the following conditions: \beginlist \item{(a)} Let $s$ be maximal such that $3^s$ divides $m$. Then $s\geq 1$; $m$ is odd; $m$ and $r$ are coprime. \item{(b)} Let $n$ be the multiplicative order of $r$ modulo $m$. Then $3$ does not divide $n$; each prime divisor of $n$ divides $m$. \item{(c)} Let $m'$ be maximal such that $m'$ divides $m$ and $m'$ is coprime to $n$. Then $r = 1$ mod $(m/m')$. \item{(d)} $k = -1$ mod $3^s$, $k = 1$ mod $(m/m')$ and $k^2 = 1$ modulo $m$. \endlist Let $p,q,a,b$ generate a type-III-group determined by $m,r$ with relations as given in Section `FpfRepresentations3'. Additionally, let $z^2 = p^2, z^{-1}pz = qp, z^{-1}qz = q^{-1}$ and $z^{-1}a z = a^k,z^{-1}b z = b$. Then the group $\Phi$ with generators $p,q,a,b$ and $z$ is of type IV and has size $16mn$. For $r \neq 1$ mod $m$, `FpfRepresentations4' returns a list of matrices $\{ (P, Q, A_i,B_i, Z_i) | i$ in $indexlist \}$ over GF(<p>) as well as the list $indexlist$. For $r = 1$ mod $m$, the function `FpfRepresentations4' returns $\{ (P, Q, A_i, Z_i) | i$ in $indexlist \}$ over GF(<p>) and $indexlist$. The GF(<p>)-representations determined by $p \mapsto P, q \mapsto Q$ and $a \mapsto A_i, b \mapsto B_i, z \mapsto Z_i$ are all irreducible, fpf representations of $\Phi$ upto equivalence. We determine the $2$ nonequivalent irreducible matrix representations of the smallest type-IV-group (binary octahedral group, m = 3, r = 1, k = -1, size 48) over the field GF(7): \beginexample gap> aux := FpfRepresentations4( 7, 3, 1, -1 ); [ [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ], [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ], [ [ Z(7)^2, 0*Z(7) ], [ Z(7)^0, Z(7)^4 ] ], [ [ Z(7)^5, Z(7) ], [ Z(7), Z(7)^2 ] ] ], [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ], [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ], [ [ Z(7)^2, 0*Z(7) ], [ Z(7)^0, Z(7)^4 ] ], [ [ Z(7)^2, Z(7)^4 ], [ Z(7)^4, Z(7)^5 ] ] ] ], [ [ 1, 1 ], [ -1, 1 ] ] ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Fixed-point-free automorphism groups} \>FpfAutomorphismGroupsCyclic( <ints>, <m> ) If `AbelianGroup(<ints>)' admits a cyclic fpf automorphism group of size <m>, then `FpfAutomorphismGroupsCyclic' determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(<ints>)'. <ints> has to be a list of prime power integers and is sorted in the function, according to the order $p^i\leq q^j \Leftrightarrow p \< q$ or ($p=q$ and $j \< i$). `AbelianGroup(<ints>)' admits a cyclic fpf automorphism group of size <m> iff the multiplicity of each prime power $p^i$ in <ints> is divisible by `DegreeOfIrredFpfRepCyclic( p, m )'. A list of generators of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(<ints>)', on which the automorphisms act. Here <ints> is sorted with the order above. The generators, <as>, of the $2$ nonconjugate cyclic fpf automorphism groups of order $4$ on $Z_{25}\times Z_{5}$ are computed as follows: \beginexample gap> aux := FpfAutomorphismGroupsCyclic( [25,5], 4 ); [ [ [ f1, f3 ] -> [ f1^2*f2, f3^2 ], [ f1, f3 ] -> [ f1^2*f2, f3^3 ] ], <pc group of size 125 with 2 generators> ] gap> as := aux[1]; [ [ f1, f3 ] -> [ f1^2*f2, f3^2 ], [ f1, f3 ] -> [ f1^2*f2, f3^3 ] ] gap> G := aux[2]; <pc group of size 125 with 2 generators> \endexample \>FpfAutomorphismGroupsMetacyclic( <ints>, <m>, <r> ) If `AbelianGroup(<ints>)' admits a metacyclic fpf automorphism group determined by parameters <m> and <r> that is not cyclic (see `FpfRepresentationsMetacyclic' for a presentation), then `FpfAutomorphismGroupsMetacyclic' determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(<ints>)'. <ints> has to be a list of prime power integers and is sorted in the function, according to the order $p^i\leq q^j \Leftrightarrow p \< q$ or $(p = q$ and $i\geq j)$. Moreover, the multiplicity of each prime power $p^i$ in <ints> has to be divisible by `DegreeOfIrredFpfRepMetacyclic( p, m, r )', which is a multiple of the multiplicative order of $r$ modulo $m$. A list of pairs of generators ($a,b$ satisfying $b^{-1}ab = a^r, a^m = 1$ and $b^n = a^{m'}$) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(<ints>)', on which the automorphisms act. Here <ints> is sorted with the order above. For $G = (Z_{3})^2\times(Z_{5})^2$ the quaternion fpf automorphism group of size $8$ (parameters $m = 4, r = -1$) is computed as follows: \beginexample gap> aux := FpfAutomorphismGroupsMetacyclic( [3,3,5,5], 4, -1 ); [ [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ], [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ], <pc group of size 225 with 4 generators> ] gap> fs := aux[1]; [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ], [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ] gap> phi := Group( fs[1] ); <group with 2 generators> gap> G := aux[2]; <pc group of size 225 with 4 generators> \endexample On $G = (Z_{7})^2\times(Z_{17})^2$ there are $2$ nonconjugate fpf automorphism groups isomorphic to the generalized quaternion group of size $16$ (parameters $m = 8, r = -1$): \beginexample gap> aux := FpfAutomorphismGroupsMetacyclic( [7,7,17,17], 8, -1 );; gap> fs := aux[1]; [ [ [ f1, f2, f3, f4 ] -> [ f1^9, f2^2, f3^4*f4^2, f3*f4^6 ], [ f1, f2, f3, f4 ] -> [ f2^16, f1, f3^4*f4^5, f3^5*f4^3 ] ], [ [ f1, f2, f3, f4 ] -> [ f1^9, f2^2, f3^3*f4^5, f3^6*f4 ], [ f1, f2, f3, f4 ] -> [ f2^16, f1, f3^3*f4^4, f3*f4^4 ] ] ] gap> phis := List( fs, Group ); [ <group with 2 generators>, <group with 2 generators> ] gap> G := aux[2]; <pc group of size 14161 with 4 generators> \endexample \>FpfAutomorphismGroups2( <ints>, <m>, <r>, <k> ) If `AbelianGroup(<ints>)' admits an fpf automorphism group of type II, determined by parameters <m>, <r>, <k> that is not metacyclic (see `FpfRepresentations2' for a presentation), then `FpfAutomorphismGroups2' determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(<ints>)'. <ints> has to be a list of prime power integers and is sorted in the function, according to the order $p^i\leq q^j \Leftrightarrow p \< q$ or $(p = q$ and $i\geq j)$. Note, that the degree of an irreducible fpf representation of a type-II-group which is not metacyclic is divisible by $4$ and that the multiplicity of each prime power $p^i$ in <ints> has to be divisible by `DegreeOfIrredFpfRep2( p, m, r, k )'. A list of triples of generators ($a,b,z$ satisfying $b^{-1}ab = a^r, a^m = 1$ and $z^{-1}az = a^{k}$) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(<ints>)', on which the automorphisms act. Here <ints> is sorted with the order above. Upto conjugacy there is only one fpf automorphism group of type II with parameters $m = 30, r = 11, k = -1$, size $120$ on the elementary abelian group of size $11^4$: \beginexample gap> aux := FpfAutomorphismGroups2( [11,11,11,11], 30, 11, -1 ); [ [ [ [ f1, f2, f3, f4 ] -> [ f1^5*f2^4, f1^3*f2^10, f3^2*f4^8, f3^6*f4 ], [ f1, f2, f3, f4 ] -> [ f1^3*f2^10, f1^10*f2^8, f3^8*f4, f3*f4^3 ], [ f1, f2, f3, f4 ] -> [ f3^10, f4^10, f1, f2 ] ] ], <pc group of size 14641 with 4 generators> ] gap> phi := Group( aux[1][1] ); <group with 3 generators> gap> G := aux[2]; <pc group of size 14641 with 4 generators> \endexample \>FpfAutomorphismGroups3( <ints>, <m>, <r> ) If `AbelianGroup(<ints>)' admits an fpf automorphism group of type III determined by parameters <m> and <r> (see `FpfRepresentations3' for a presentation), then `FpfAutomorphismGroups3' determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(<ints>)'. <ints> has to be a list of prime power integers and is sorted in the function, according to the order $p^i\leq q^j \Leftrightarrow p \< q$ or $(p = q$ and $i\geq j)$. Moreover, the multiplicity of each prime power $p^i$ in <ints> has to be divisible by `DegreeOfIrredFpfRep3( p, m, r )', which is a multiple of $2n$ where $n$ is the multiplicative order of $r$ modulo $m$. A list of tuples of generators, <[p,q,a,b]>, ($p,q$ generating the quaternion group, $a,b$ satisfying $b^{-1}ab = a^r, a^m = 1$ and $b^n = a^{m'}$) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(<ints>)', on which the automorphisms act. Here <ints> is sorted with the order above. For $G = (Z_{5})^2$ the fpf automorphism type-III-group isomorphic to SL(2,3) is computed as follows (parameters $m = 3, r = 1$): \beginexample gap> aux := FpfAutomorphismGroups3( [5,5], 3, 1 ); [ [ [ [ f1, f2 ] -> [ f1^2, f2^3 ], [ f1, f2 ] -> [ f2^4, f1 ], [ f1, f2 ] -> [ f1^3*f2, f1^2*f2 ] ] ], <pc group of size 25 with 2 generators> ] gap> phi := Group( aux[1][1] ); <group with 3 generators> gap> G := aux[2]; <pc group of size 25 with 2 generators> \endexample \>FpfAutomorphismGroups4( <ints>, <m>, <r>, <k> ) If `AbelianGroup(<ints>)' admits an fpf automorphism group of type IV determined by parameters <m>, <r>, <k> (see `FpfRepresentations4' for a presentation), then `FpfAutomorphismGroups4' determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(<ints>)'. <ints> has to be a list of prime power integers and is sorted in the function, according to the order $p^i\leq q^j \Leftrightarrow p \< q$ or $(p = q$ and $i\geq j)$. Moreover, the multiplicity of each prime power $p^i$ in <ints> has to be divisible by `DegreeOfIrredFpfRep4( p, m, r )', which is a multiple of $2n$ where $n$ is the multiplicative order of $r$ modulo $m$. A list of tuples of generators, <[p,q,a,b,z]>, of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(<ints>)', on which the automorphisms act. Here <ints> is sorted with the order above. If $r = 1$ mod $m$, then a list of tuples, <[p,q,a,z]>, is returned instead. For $G = (Z_{7})^2$ the fpf automorphism type-IV-group isomorphic the binary octahedral group of size 48 (parameters $m = 3, r = 1, k = -1$) is computed as follows: \beginexample gap> aux := FpfAutomorphismGroups4( [7,7], 3, 1, -1 ); [ [ [ [ f1, f2 ] -> [ f1^2*f2^3, f1^3*f2^5 ], [ f1, f2 ] -> [ f2^6, f1 ], [ f1, f2 ] -> [ f1^2, f1*f2^4 ], [ f1, f2 ] -> [ f1^5*f2^3, f1^3*f2^2 ] ] ], <pc group of size 49 with 2 generators> ] gap> phi := Group( aux[1][1] ); <group with 4 generators> gap> G := aux[2]; <pc group of size 49 with 2 generators> \endexample %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% TeX-master: t %%% End: